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=
=−
With
λ
2wehave
z
x
from the 1st equation. Substituting this in the 2nd
equation we have
y
=
4
x
, which permits us to state that the associated eigenvector
T
.
is of the form
[
k
4
k
−
k
]
0 from the 3rd equation,
which permits us to state that the associated eigenvector is of the form
With
λ
=
3wehave
z
=
0 from the 1st equation, and
x
=
T
.
[
0
k
0
]
With
λ
=
4wehave
z
=
x
from the 1st equation. Substituting this in the 2nd
equation we have
y
=
2
x
, which permits us to state that the associated eigenvector
T
.
Therefore, the eigenvectors and eigenvalues are
is of the form
[
k
2
kk
]
T
[
k
4
k
−
k
]
λ
=
2
T
[
0
k
0
]
λ
=
3
T
[
k
2
kk
]
λ
=
4
where
k
0.
The major problem with the above technique is that it requires careful analysis to
untangle the eigenvector, and ideally, we require a deterministic algorithm to reveal
the result. We will discover that such a technique is available in Chap. 9.
=
4.18 Vector Products
Vectors are regarded as single column or single row matrices, which permits us to
express their products neatly. For example, given two vectors
⎡
⎤
⎡
⎤
a
b
c
x
y
z
⎣
⎦
,
⎣
⎦
v
=
w
=
then
v
T
w
=
abc
v
·
w
=
⎡
⎣
⎤
⎦
=
x
y
z
ax
+
by
+
cz.
Similarly, the vector cross product is written
⎡
⎤
⎡
⎤
⎡
⎤
a
b
c
x
y
z
i jk
abc
xyz
⎣
⎦
×
⎣
⎦
=
⎣
⎦
v
×
w
=
=
(bz
−
cy)
i
−
(az
−
xc)
j
+
(ay
−
bx)
k
⎡
⎣
⎤
⎦
.
−
bz
cy
=
−
az
+
xc
ay
−
bx
